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We show that for a generic polynomial $H=H(x,y)$ and an arbitrary differential 1-form $\omega =P(x,y)dx+Q(x,y)dy$ with polynomial coefficients of degree $\le d$, the number of ovals of the foliation $H=\mathrm{const}$, which yield the zero value of the complete Abelian integral $I\left(t\right)={\oint }_{H=t}\omega$, grows at most as $exp{O}_H\left(d\right)$ as $d\to \infty$, where $O_H\left(d\right)$ depends only on $H$. The main result of the paper is derived from the following more general theorem on bounds for isolated zeros occurring in polynomial envelopes of linear differential equations. Let $f_1\left(t\right),\cdots ,f_n\left(t\right)$, $t\in K⋐ℝ$, be a fundamental system of real solutions...

We give an explicit upper bound for the number of isolated intersections between an integral curve of a polynomial vector field in ${ℝ}^n$ and an algebraic hypersurface. The answer is polynomial in the height (the magnitude of coefficients) of the equation and the size of the curve in the space-time, with the exponent depending only on the degree and the dimension. The problem turns out to be closely related to finding an explicit upper bound for the length of ascending chains of polynomial...